Existence results for classes of infinite semipositone problems
© Goddard II et al.; licensee Springer. 2013
Received: 23 October 2012
Accepted: 5 April 2013
Published: 19 April 2013
We consider the problem
where , , Ω is a smooth bounded domain in , , , , and . Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case is presented.
where Ω is a smooth bounded domain in , , , , is the Laplacian of u and is a function satisfying for , , and for . Existence of positive solutions of problem (1) was studied in . In particular, it was proved that given an and there exists a such that for (1) has positive solutions. Here, is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when . Later in , these results were extended to the case of the p-Laplacian operator, , where , . Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity satisfies for some . See [3–9] for some existence results for semipositone problems.
where , , Ω is a smooth bounded domain in , , , , , , and . In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity satisfies . One can refer to [10–14], and [15–17] for some recent existence results of infinite semipositone problems. We establish the following theorem.
Theorem 1.1 Given , , and , there exists a constant such that for , (2) has a positive solution.
Remark 1.1 In the nonsingular case (), positive solutions exist only when (the principal eigenvalue) (see [1, 2]). But in the singular case, we establish the existence of a positive solution for any given .
where . We assume:
() and satisfies for , and for some θ such that .
We note that if then is nonsingular at 0 and . In this case, problem (4) can be studied using ideas in the proof of Theorem 1.1. Hence, our focus is on the case when in which, h may be singular at 0. Note that in this case .
Remark 1.2 Note that since .
We then establish the following theorem.
Theorem 1.2 Given , , , and assume () holds. Then there exists a constant such that for , (3) has a positive radial solution.
where Ω is a smooth bounded domain in , a is a positive parameter, , and . We prove the following.
where . The following lemma was established in .
Let ψ be a subsolution of (2) and Z be a supersolution of (2) such that in Ω. Then (2) has a solution u such that in Ω.
Finding a positive subsolution, ψ, for such infinite semipositone problems is quite challenging since we need to construct ψ in such a way that and in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form , where k is an appropriate positive constant, and is the eigenfunction corresponding to the first eigenvalue of in Ω, on ∂ Ω.
In Sections 2, 3, and 4, we provide proofs of our results. Section 5 is concerned with providing some exact bifurcation diagrams of positive solutions of (2) when and .
2 Proof of Theorem 1.1
From (11), (12) and (13) we see that equation (7) holds in Ω, if . Next, we construct a supersolution. Let e be the solution of in on ∂ Ω. Choose such that and . Define . Then Z is a supersolution of (2). Thus, Theorem 1.1 is proven.
3 Proof of Theorem 1.2
Proving (18) holds in is straightforward since h is not singular at . Thus, from equations (19), (20) and (21), we see that (15) holds in . Hence, ψ is a subsolution. Let where e satisfies in , and is such that and . Then Z is a supersolution of (4) and there exists a solution u of (4) such that . Thus, Theorem 1.2 is proven.
4 Proof of Theorem 1.3
Thus, ψ is a subsolution. It is easy to see that is a supersolution of (6). Since k, can be chosen small enough, . Thus, (6) has a positive solution for every . Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions at .
5 Numerical results
EK Lee was supported by 2-year Research Grant of Pusan National University.
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